Question

The following measurements of$\mathit{x}$ and $\mathit{y}$have been made.

$\begin{array}{l}\mathbf{x}\mathbf{:}\mathbf{5.1}\mathbf{,}\mathbf{4.9}\mathbf{,}\mathbf{5.0}\mathbf{,}\mathbf{5.2}\mathbf{,}\mathbf{4.9}\mathbf{,}\mathbf{5.0}\mathbf{,}\mathbf{4.8}\mathbf{,}\mathbf{5.1}\\ \mathbf{y}\mathbf{:}\mathbf{1.03}\mathbf{,}\mathbf{1.05}\mathbf{,}\mathbf{0.96}\mathbf{,}\mathbf{1.00}\mathbf{,}\mathbf{1.02}\mathbf{,}\mathbf{0.95}\mathbf{,}\mathbf{0.99}\mathbf{,}\mathbf{1.01}\mathbf{,}\mathbf{1.00}\mathbf{,}\mathbf{0.99}\end{array}$

Find the mean value and the probable error of$\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{x}\mathbf{+}\mathit{y}\mathbf{,}\mathit{x}\mathit{y}\mathbf{,}{\mathit{x}}^{\mathbf{3}}\mathit{s}\mathit{i}\mathit{n}\mathit{y}$and.$\mathbf{ln}\mathbf{\left(}\mathit{x}\mathbf{\right)}$

Short answer

Required answersare,

$\begin{array}{c}\overline{x}=5,\\ r_x=0.031\\ \overline{y}=1,\\ r_y=0.0064\end{array}$

$\begin{array}{c}\overline{w}=6,\\ r_w=0.0317\\ \overline{xy}=5,\\ r_{xy}=0.044\end{array}$

$\begin{array}{c}\overline{x\sin \left(y\right)}=105,\\ r=2\\ \overline{\ln x}=1.609,\\ r=0.006\end{array}$

Step-by-step solution

Given Information

Measurements are mentioned below:

$\begin{array}{l}x:5.1,4.9,5.0,5.2,4.9,5.0,4.8,5.1\\ y:1.03,1.05,0.96,1.00,1.02,0.95,0.99,1.01,1.00,0.99\end{array}$

Definition of Probable error. 

The amount by which a sample's arithmetic mean is predicted to vary solely due to chance.

Calculate mean and probable error value of  . 

Calculate expected value.

$\begin{array}{c}\overline{x}=\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{n}\\ =\frac{2\times 5.1+2\times 4.9+2\times 5.0+5.2+4.8}{8}\\ =5\end{array}$

Calculate variance.

$\begin{array}{c}\mathrm{Var}\left(x\right)=\frac{1}{n_x-1}\sum _{i=1}^n{(x_i-\overline{x})}^2\\ =\frac{1}{7}[4\times {\left(0.1\right)}^2+2\times {\left(0.2\right)}^2]\\ =\frac{3}{175}\end{array}$

Calculate standard deviation.

$\begin{array}{c}{\sigma }_x=\sqrt{\mathrm{Var}\left(x\right)}\\ =\sqrt{\frac{3}{175}}\\ =0.13\end{array}$

Calculate error of values.

$\begin{array}{c}{\sigma }_{mx}=\sqrt{\frac{\mathrm{Var}\left(x\right)}{n}}\\ =\sqrt{\frac{3}{175\times 8}}\\ =0.046\end{array}$

Calculate probable error.

$\begin{array}{c}{r}_x=0.6745\left(0.046\right)\\ =0.031\end{array}$

Calculate mean and probable error value of .y

Calculate expected value.

$\begin{array}{c}\overline{y}=\frac{{\displaystyle \sum _{i=1}^n{y}_i}}{n}\\ =\frac{2\times 1.00+2\times 0.99+1.03+1.05+1.02+0.95+1.01+0.96}{10}\\ =1\end{array}$

Calculate variance.

$\begin{array}{c}{\sigma }_y^2=\frac{1}{n_y-1}\sum _{i=1}^n{(y_i-\overline{y})}^2\\ =\frac{1}{9}[{\left(0.03\right)}^2+2{\left(0.05\right)}^2+3{\left(0.01\right)}^2+{\left(0.02\right)}^2+{\left(0.04\right)}^2]\\ =\frac{41}{45000}\end{array}$

Calculate standard deviation.

$\begin{array}{c}{\sigma }_y=\sqrt{\mathrm{Var}\left(y\right)}\\ =\sqrt{\frac{41}{45000}}\\ =0.03\end{array}$

Calculate error of values.

$\begin{array}{c}{\sigma }_{my}=\sqrt{\frac{\mathrm{Var}\left(y\right)}{n_y}}\\ =\sqrt{\frac{41}{45000\times 10}}\\ =0.0095\end{array}$

Calculate probable error.

$\begin{array}{c}{r}_y=0.6745\left(0.0095\right)\\ =0.0064\end{array}$

Calculate mean and probable error value of.x+y

Calculate expected value.

$\begin{array}{c}\overline{w}=\overline{x+y}\\ =\overline{x}+\overline{y}\\ =5+1\\ =6\end{array}$

Calculate error of values.

$\begin{array}{c}{\sigma }_{mw}=\sqrt{{\sigma }_{my}^2+{\sigma }_{mw}^2}\\ =\sqrt{{\left(0.0095\right)}^2+{\left(0.046\right)}^2}\\ =0.047\end{array}$

Calculate probable error.

$\begin{array}{c}{r}_{\text{w}}=0.6745\left(0.047\right)\\ =0.0317\end{array}$

Calculate mean and probable error valueof  xy.

Calculate expected value.

$\begin{array}{c}w=xy\\ E\left(w\right)={\mu }_x{\mu }_y\\ =5\times 1\\ =5\end{array}$

Calculate error of values.

$\begin{array}{c}{\sigma }_{mw}={\left[\sqrt{{\left(\frac{\partial w}{\partial x}\right)}^2{\sigma }_{mx}^2+{\left(\frac{\partial w}{\partial y}\right)}^2{\sigma }_{my}^2}\right]}_{(x,y)=(x,y)}\\ {\sigma }_{mw}=\sqrt{{\left(1\right)}^2{\left(0.046\right)}^2+{\left(5\right)}^2{\left(0.0095\right)}^2}\\ =0.066\end{array}$

Calculate probable error.

$\begin{array}{c}{r}_{\text{w}}=\left(0.6745\right)\left(0.066\right)\\ =0.044\end{array}$

Calculate mean and probable error value of.x3sin(y) 

Calculate expected value.

$\begin{array}{c}f(x,y)=x^3\sin \left(y\right)\\ E\left(f\right)={\mu }_x^3\sin \left({\mu }_y\right)\\ =5^3\sin \left(1\right)\\ =105\end{array}$

Calculate error of values.

$\begin{array}{c}{\sigma }_{mf}=\sqrt{{\left[3x^2\sin \left(y\right)\right]}^2{\sigma }_{\mathrm{me}}^2+{\left[x^3\cos \left(y\right)\right]}^2{\sigma }_{\text{my}}^2}\\ =\sqrt{{[3\times 25\sin (1\left)\right]}^2{\left[0.046\right]}^2+{[125\times \cos (1\left)\right]}^2{\left(0.0995\right)}^2}\\ =3\end{array}$

Calculate probable error.

$\begin{array}{c}{r}_f=\left(3\right)\left(0.6745\right)\\ =2\end{array}$

Calculate mean and probable error value of.ln(x)

Calculate expected value.

$\begin{array}{c}E\left(z\right)=\ln \left({\mu }_s\right)\\ =\ln \left(5\right)\\ =1.609\end{array}$

Calculate error of values.

$\begin{array}{c}{\sigma }_{ms}=\sqrt{\frac{1}{25}\times {\left(0.046\right)}^2}\\ =\frac{23}{2500}\end{array}$

Calculate probable error.

$\begin{array}{c}{r}_z=0.6745\left(\frac{23}{2500}\right)\\ =0.006\end{array}$

Hence, required expressions are,

$\begin{array}{c}\overline{x}=5,\\ r_x=0.031\\ \overline{y}=1,\\ r_y=0.0064\end{array}$

$\begin{array}{c}\overline{w}=6,\\ r_w=0.0317\\ \overline{xy}=5,\\ r_{xy}=0.044\end{array}$

$\begin{array}{c}\overline{x\sin \left(y\right)}=105,\\ r=2\\ \overline{\ln x}=1.609,\\ r=0.006\end{array}$